Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem

“…Estimate (39) is open even when n = 2 however; the known partial results on L 4 (R 2 ) correspond to loss of 1 8 derivatives ( [36]; an improvement to loss of 1 8 − ǫ derivatives appears implicit in [12], p. 60). In general dimensions, the following implications are known.…”

Section: Oscillatory Integrals and Kakeyamentioning

confidence: 99%

Recent work connected with the Kakeya problem

Wolff

2003

Lectures on Harmonic Analysis

200

275

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“…We obtain a result for truncated cones by combining a square function estimate with a result for the Nikodym maximal function for light rays. For the square function estimate we can adapt Bourgain's argument [1] or in R 2 × R there is a result of Mockenhaupt, Seeger and Sogge (MSS) [13]. Mockenhaupt, Seeger and Sogge [14] and Skarabot [17] have determined maximal function results, and in R 2 × R there is a maximal function bound due to Córdoba [5].…”

Section: Bochner-riesz Means 1413mentioning

confidence: 99%

Bochner-Riesz means with respect to a rough distance function

Taylor

2006

Trans. Amer. Math. Soc.

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“…These problems can be regarded as special cases of the local smoothing problem for the wave equation [10], [8] -a slightly weaker form of (3) (namely: The same estimate with the L 3 norm replaced by the L 4 norm on both sides of the inequality) would follow readily from the "sharp local smoothing" conjecture made in [10], [8].…”

mentioning

confidence: 99%

On some variants of the Kakeya problem

Kolasa¹,

Wolff²

1999

Pacific J. Math.

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We study the question of lower bounds for the Hausdorff dimension of a set in R n containing spheres of every radius. If n ≥ 3 then such a set must have dimension n. If n = 2 then it must have dimension at least 11/6. We also study the analogous maximal function problem and related problem of Besicovitch sets with an axis of symmetry.Besicovitch and Rado [1] and Kinney [5] proved the following result: There is a closed set E ⊂ R 2 with measure zero which contains a circle of every radius.The construction of Besicovitch and Rado works in R d : If d ≥ 3 there is a closed set E ⊂ R d with measure zero which contains a sphere of every radius. We will give an exposition of this construction in Section 1 below.One can ask whether a set containing a sphere of every radius must have Hausdorff dimension d. As it turns out, this question is easily answered in higher dimensions. In R 2 this may still be true but appears harder. One purpose of this paper is to prove the following partial result.Theorem 2. A Borel set in R 2 which contains a circle of every radius has Hausdorff dimension at least 11 6 . Following a known pattern (see [2] for example) we will derive Theorems 1 and 2 from L p → L q estimates for a related maximal function. Fix δ > 0.C δ (x, r) = {z ∈ R d : r − δ < |z − x| < r + δ}.

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Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem

Cited by 143 publications

References 16 publications

Recent work connected with the Kakeya problem

Recent work connected with the Kakeya problem

Bochner-Riesz means with respect to a rough distance function

On some variants of the Kakeya problem

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